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# Optimal Volume-Sensitive Bounds for Polytope Approximation

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LIPIcs.SoCG.2023.9.pdf
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## Acknowledgements

The authors would like to acknowledge the insights and feedback from Rahul Arya and Guilherme da Fonseca.

## Cite As

Sunil Arya and David M. Mount. Optimal Volume-Sensitive Bounds for Polytope Approximation. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.9

## Abstract

Approximating convex bodies is a fundamental question in geometry and has a wide variety of applications. Consider a convex body K of diameter Δ in ℝ^d for fixed d. The objective is to minimize the number of vertices (alternatively, the number of facets) of an approximating polytope for a given Hausdorff error ε. It is known from classical results of Dudley (1974) and Bronshteyn and Ivanov (1976) that Θ((Δ/ε)^{(d-1)/2}) vertices (alternatively, facets) are both necessary and sufficient. While this bound is tight in the worst case, that of Euclidean balls, it is far from optimal for skinny convex bodies. A natural way to characterize a convex object’s skinniness is in terms of its relationship to the Euclidean ball. Given a convex body K, define its volume diameter Δ_d to be the diameter of a Euclidean ball of the same volume as K, and define its surface diameter Δ_{d-1} analogously for surface area. It follows from generalizations of the isoperimetric inequality that Δ ≥ Δ_{d-1} ≥ Δ_d. Arya, da Fonseca, and Mount (SoCG 2012) demonstrated that the diameter-based bound could be made surface-area sensitive, improving the above bound to O((Δ_{d-1}/ε)^{(d-1)/2}). In this paper, we strengthen this by proving the existence of an approximation with O((Δ_d/ε)^{(d-1)/2}) facets. This improvement is a result of the combination of a number of new ideas. As in prior work, we exploit properties of the original body and its polar dual. In order to obtain a volume-sensitive bound, we explore the following more general problem. Given two convex bodies, one nested within the other, find a low-complexity convex polytope that is sandwiched between them. We show that this problem can be reduced to a covering problem involving a natural intermediate body based on the harmonic mean. Our proof relies on a geometric analysis of a relative notion of fatness involving these bodies.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
##### Keywords
• Approximation algorithms
• convexity
• Macbeath regions

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## References

1. P. K. Agarwal, S. Har-Peled, and K. R. Varadarajan. Geometric approximation via coresets. In J. E. Goodman, J. Pach, and E. Welzl, editors, Combinatorial and Computational Geometry. MSRI Publications, 2005.
2. R. Arya, S. Arya, G. D. da Fonseca, and D. M. Mount. Optimal bound on the combinatorial complexity of approximating polytopes. ACM Trans. Algorithms, 18:1-29, 2022. URL: https://doi.org/10.1145/3559106.
3. S. Arya and T. M. Chan. Better ε-dependencies for offline approximate nearest neighbor search, Euclidean minimum spanning trees, and ε-kernels. In Proc. 30th Annu. Sympos. Comput. Geom., pages 416-425, 2014. URL: https://doi.org/10.1145/2582112.2582161.
4. S. Arya, G. D. da Fonseca, and D. M. Mount. Optimal area-sensitive bounds for polytope approximation. In Proc. 28th Annu. Sympos. Comput. Geom., pages 363-372, 2012. URL: https://doi.org/10.1145/2261250.2261305.
5. S. Arya, G. D. da Fonseca, and D. M. Mount. Near-optimal ε-kernel construction and related problems. In Proc. 33rd Internat. Sympos. Comput. Geom., pages 10:1-15, 2017. URL: https://doi.org/10.4230/LIPIcs.SoCG.2017.10.
6. S. Arya, G. D. da Fonseca, and D. M. Mount. Economical convex coverings and applications. In Proc. 34th Annu. ACM-SIAM Sympos. Discrete Algorithms, pages 1834-1861, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch70.
7. S. Arya, G. D. da Fonseca, and D. M. Mount. Economical convex coverings and applications, 2023. URL: https://arxiv.org/abs/2303.08349.
8. K. Böröczky Jr. Approximation of general smooth convex bodies. Adv. Math., 153:325-341, 2000. URL: https://doi.org/10.1006/aima.1999.1904.
9. J. Bourgain and V. D. Milman. New volume ratio properties for convex symmetric bodies. Invent. Math., 88:319-340, 1987. URL: https://doi.org/10.1007/BF01388911.
10. E. M. Bronshteyn and L. D. Ivanov. The approximation of convex sets by polyhedra. Siberian Math. J., 16:852-853, 1976. URL: https://doi.org/10.1007/BF00967115.
11. E. M. Bronstein. Approximation of convex sets by polytopes. J. Math. Sci., 153(6):727-762, 2008. URL: https://doi.org/10.1007/s10958-008-9144-x.
12. K. L. Clarkson. Building triangulations using ε-nets. In Proc. 38th Annu. ACM Sympos. Theory Comput., pages 326-335, 2006. URL: https://doi.org/10.1145/1132516.1132564.
13. R. M. Dudley. Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory, 10(3):227-236, 1974. URL: https://doi.org/10.1016/0021-9045(74)90120-8.
14. H. G. Eggleston. Convexity. Cambridge University Press, 1958. URL: https://doi.org/10.1017/CBO9780511566172.
15. W. J. Firey. Polar means of convex bodies and a dual to the Brunn-Minkowski theorem. Canad. J. Math, 13:444-453, 1961. URL: https://doi.org/10.4153/CJM-1961-037-0.
16. P. M. Gruber. Aspects of approximation of convex bodies. In P. M. Gruber and J. M. Wills, editors, Handbook of Convex Geometry, chapter 1.10, pages 319-345. North-Holland, 1993. URL: https://doi.org/10.1016/B978-0-444-89596-7.50015-8.
17. B. Grünbaum. Measures of symmetry for convex sets. In Proc. Sympos. Pure Math., volume VII, pages 233-270, 1963. URL: https://doi.org/10.1090/pspum/007/0156259.
18. G. Kuperberg. From the Mahler conjecture to Gauss linking integrals. Geom. Funct. Anal., 18:870-892, 2008. URL: https://doi.org/10.1007/s00039-008-0669-4.
19. D. E. McClure and R. A. Vitalie. Polygonal approximation of plane convex bodies. J. Math. Anal. Appl., 51:326-358, 1975. URL: https://doi.org/10.1016/0022-247X(75)90125-0.
20. P. McMullen. Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc. Cambridge Philos. Soc, 78(2):247-261, 1975. URL: https://doi.org/10.1017/S0305004100051665.
21. P. McMullen. Inequalities between intrinsic volumes. Monatshefte für Mathematik, 111:47-53, 1991. URL: https://doi.org/10.1007/BF01299276.
22. M. Meyer and A. Pajor. On the Blaschke-Santaló inequality. Arch. Math., 55:82-93, 1990. URL: https://doi.org/10.1007/BF01199119.
23. V. D. Milman and A. Pajor. Entropy and asymptotic geometry of non-symmetric convex bodies. Adv. Math., 152:314-335, 2000. URL: https://doi.org/10.1006/aima.1999.1903.
24. H. Minkowski. Allgemeine Lehrsätze über die konvexen Polyeder. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Math-Phys. Kl., pages 198-219, 1897. URL: http://eudml.org/doc/58391.
25. F. Nazarov. The Hörmander proof of the Bourgain-Milman theorem. In Geometric Aspects of Functional Analysis, pages 335-343. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-29849-3_20.
26. J. Radon. Über eine Erweiterung des Begriffs der konvexen Functionen mit einer Anwendung auf die Theorie der konvexen Körper. S.-B. Akad. Wiss. Wien, 125:241-258, 1916.
27. J. Richter-Gebert. Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-17286-1.
28. L. A. Santaló. An affine invariant for convex bodies of n-dimensional space. Port. Math., 8:155-161, 1949. (In Spanish).
29. R. Schneider. Polyhedral approximation of smooth convex bodies. J. Math. Anal. Appl., 128:470-474, 1987. URL: https://doi.org/10.1016/0022-247X(87)90197-1.
30. R. Schneider. Convex bodies: The Brunn-Minkowski theory. Cambridge University Press, 1993. URL: https://doi.org/10.1017/CBO9780511526282.
31. G. Toth. Measures of symmetry for convex sets and stability. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-23733-6.
32. L. F. Toth. Approximation by polygons and polyhedra. Bull. Amer. Math. Soc., 54:431-438, 1948.
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